I'm assuming someone you know is attempting to obtain a security clearance, and has been told that they are (or will be) ineligible unless they renounce their Italian citizenship?

I would think that they would need to contact an Italian embassy/consulate in order to officially renounce. They would then probably be required to provide some sort of official documentation that the process had been completed in order to be reconsidered for the security clearance. This is probably done by making some sort of declaration in front of an official at an Italian embassy or consulate, or maybe by submitting something in writing.

I am not certain that this is the actual procedure to follow, but it does deal with security clearances and renouncing citizenship (found on another site dealing with the same topic):

(5) Dual citizenship. Applicants who hold dualcitizenship may not be eligible for a securityclearance. Those who cannot receive a securityclearance cannot hold an AR appointment.Therefore all applicants who hold dualcitizenship must be willing to renounce their non-U.S. citizenship. U.S. citizens with dual citizenshipand/or foreign passport must providedocumentation renouncing foreign citizenshipand relinquishing the foreign passport as anappointment eligibility requirement. They areineligible for HPSP entry until this requirement ismet. Procedures for renouncing non-U.S. citizenship:(a) Applicant signs a notarized letter addressedto appropriate embassy relinquishingcitizenship and enclosing passport.(b) Applicant provides HCR copy of the notarizedletter to embassy and Federal Expressreceipt or proof from embassy that passportwas relinquished to embassy.(c) HCR sends copy of Federal Express receiptand notarized letter or proof from embassythat passport has been relinquished to embassywith EPSQ in HPSP application.(d) HCR sends copy of Federal Express receiptand notarized letter to embassy with EPSQ.USAREC Regulation 601-105 Health Professions Scholarship Program Instruction Handbook on Applicant and Selectee Processing"